Schulze Method - Simpler Definition

Norman Petry npetry at
Tue Sep 8 13:09:10 PDT 1998

Markus has suggested a couple of small clarifications to my definition of
his method which I thought I'd pass along:

>I want to propose two changes in the definition of the
>Schulze method.
>First: It should be clarified, that whenever a ballot is
>chosen randomly it is chosen among those ballots that
>haven't yet been chosen randomly.
>The reason: Someone could argue, that it could happen, that
>always the same ballots are chosen randomly, so that the
>Schulze method doesn't lead to a result in a finite amount
>of time.
>Second: It should be clarified, that whenever every ballot
>has been chosen randomly and there is still more than one
>winner, then the winner is chosen randomly among the
>potential winners.

One way to incorporate these changes in a description of the _complete_
Schulze method is to use the notation that was developed on this list some
time ago for compound methods.  For example:


meant the Smith method followed by the "Condorcet" method, where Condorcet's
method in this case was intended to mean the candidate with the fewest
votes-against in his/her worst pairwise loss (this is sometimes called the
"Minimax" rule).  It's understood using this notation that when one stage in
the compound method results in a tie, the tied subset of potential winners
are run through the overall method again, until a single-winner is found.

Using this notation, the complete Schulze method would be:

Schulze's Method = Beat-Path Undefeated//Random Ballot//Random.

with "Beat-Path Undefeated" as the core of the method.  Alternately, if
there are a large number of candidates, the Smith set could be used as the
initial filter, since Smith is easier to compute than Beat-Path Unbeaten.
In that case, the method is:

Schulze's Method = Smith//Beat-Path Undefeated//Random Ballot//Random.

In any case, the two definitions are equivalent, since Beat-Path Undefeated
will never select winners outside the Smith set anyway.


Finally, some (obvious) definitions:

Random Ballot:

Select a random ballot and remove it from the election.  The winners are the
candidates ranked highest on this ballot who have not yet been eliminated.


Select the winner randomly from candidates not yet eliminated.


Norm Petry

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